ZEF-Discussion Papers on
Development Policy No. 168
Oded Stark and Marcin Jakubek
Migration networks as a response to
financial constraints: Onset and
endogenous dynamics
Bonn, August 2012
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DEVELOPMENT POLICY include Joachim von Braun (Chair), Solvey Gerke, and Manfred
Denich.
Oded Stark and Marcin Jakubek, Migration networks as a response to financial
constraints: Onset and endogenous dynamics, ZEF - Discussion Papers on Development
Policy No. 168, Center for Development Research, Bonn, August 2012, pp. 22.
ISSN: 1436-9931
Published by:
Zentrum für Entwicklungsforschung (ZEF)
Center for Development Research
Walter-Flex-Straße 3
D – 53113 Bonn
Germany
Phone: +49-228-73-1861
Fax: +49-228-73-1869
E-Mail: zef@uni-bonn.de
www.zef.de
The authors:
Oded Stark, University of Bonn. Contact: ostark@uni-bonn.de
Marcin Jakubek, University of Klagenfurt. Contact: MarcinKacper.Jakubek@uni-klu.ac.at
Acknowledgements
We are indebted to Kaivan Munshi for advice and guidance, and to anonymous referees,
Agnieszka Dorn, Gerhard Sorger, and Yong Wang for enlightening comments.
Abstract
A migration network is modeled as a mutually beneficial cooperative agreement between
financially-constrained individuals who seek to finance and expedite their migration. The
cooperation agreement creates a network: “established” migrants contract to support the
subsequent migration of others in exchange for receiving support themselves. When the
model is expanded to study cooperation between more than two migrants, it emerges that
there is a finite optimal size of the migration network. Consequently, would-be migrants in
the sending country will form a multitude of networks, rather than a single grand network.
Keywords: Migration networks; Schedule of migration; Sequential migration; Affinity;
Interpersonal bonds; Cost of migration
JEL classification: D01; D71; D90; F22; F24; J61; O15
1. Introduction
Migration in general, and migration in developing countries in particular, is rarely an isolated
event, and is nearly always a sequence of moves - a process in which earlier migrants shape
the migration infrastructure of today’s would-be migrants. The intertemporal linkages can
and often do assume the form of a migration network. In this paper we develop the idea that
the phased nature of migration is caused by the endogenous dynamics of the operation of
migration networks, and that a migration network evolves as a response to financial
constraints. Specifically, we model a migration network as an arrangement between
financially-constrained individuals who, in a manner akin to the functioning of a Rotating
Savings and Credit Association (ROSCA), seek to finance and expedite their migration.
Thus, we combine two strands of the literature, allowing us to view a migration network as
an informal financial cooperation scheme that spans time and space.
Research on networks as facilitators of migration has shown that network-type links
account for a single migration turning into a migration process, as would-be migrants tread
the path chartered by others. Myrdal (1957) drew attention to the power and role of
cumulative causation - the self-perpetuating interplay between networks that encourages
additional migration, which, in turn, reinforces the network itself, causing it to grow and
become more efficient in helping other would-be migrants. Taylor (1986) shows that
networks play a crucial role in the evolution of migration, especially in the dynamics of
international migration, where migration risks are highest, labor market information is most
costly and scarce, and the penalty for making bad forecasts is most severe. Networks
influence both the direction and the magnitude of migration over time. The network effect is
strongest when a member of a single village household establishes himself at a particular
destination, and less strong when those concerned come from other village households.
Massey (1990) notes that the social capital of migrant networks lowers the costs and risks
associated with migration, thereby raising the net benefit from migration. A large body of
empirical work shows that the cross-border links that migration networks provide have a
significant positive impact on the intensity (rate) of migration (Davis and Winters, 2001;
Dolfin and Genicot, 2010). Orrenius and Zavodny (2005) find that the likelihood of a young
Mexican male migrating to the U.S. is positively correlated with his father having migrated
and with the number of siblings who have migrated. Hanson and McIntosh (2010) document
how, to some extent, networks act as substitutes for a wage differential in moving the
migration flow between Mexico and the U.S. in the period 1960 to 2000. Beaman (2012)
1
looks at how within-network competition for job information could weaken the effectiveness
of a network as a device that overcomes labor market imperfection, and assesses the
relationship between the size of the network and its effectiveness. Although the empirical
context of her work (refugees in the U.S.) is distinct from ours, the perspectives of her
research, namely the inner composition of the network and its optimal size, are akin to ours.
Massey (1990) defines migration networks as “sets of interpersonal ties that link migrants,
former migrants, and nonmigrants in origin and destination areas by the bonds of kinship,
friendship, and shared community origin.” We model the intensity of interpersonal bonds
(affinity) and we identify the precise role that such bonds play in the design of a network.
Often, the support provided by the “network” is critical to subsequent migration;
without that support, follow-up migration will not take place. What is the underlying
rationale for providing support? Even though it is not hard to see why would-be migrants
accept assistance from established migrants, what prompts the latter to provide assistance?
And could it be that the first act in establishing a “network” actually takes place at origin
rather than at destination?
Given the role that networks play, it is somewhat surprising that there has been no
formal economic theory of migration networks. In this paper we take a step towards
correcting this lacuna. We ask: why are migration networks formed? In what circumstances
are networks more likely to emerge or evolve? Under what conditions will individuals join
networks? What benefit does belonging to a network confer compared with “going it alone?”
What determines the (optimal) size of a network? What constrains this size?
We model migration network as a form of cooperation between financiallyconstrained would-be migrants aimed at shortening the time required to accumulate the
resources needed to pay for the cost of migration and initial settlement in the country of
destination. 1 Seen this way, a migration network is a mutually beneficial cooperative
arrangement
between
financially-constrained,
utility-maximizing
individuals,
an
implementation of an exchange arrangement that binds individuals across the sending and
receiving countries and over time. This perspective complements the view of migration
networks as conveyors of information, especially about job opportunities, and as suppliers of
1
There is a perception in the migration literature that the cost of migration to the n-th individual, including the
cost of getting established at destination, is not independent of the presence at destination of past n-1 migrants.
The standard argument in the received literature (c.f., for example, Carrington et al., 1996) is that the cost
decreases in n-1. But this is not what interests us. We study the case in which the overall cost is given, and we
show how cost sharing is arranged such that established migrants bear part of the cost.
2
a variety of types of support with which established migrants furnish would-be and newlyarriving migrants. 2 Moreover, in the received literature, the emergence and formation of
migration networks are typically not explained; rather, their role is highlighted. For example,
Hanson and McIntosh (2010) refer to networks as “pre-existing” or “historical,” and
Carrington et al. (1996) relate to migrant networks as “self-perpetuating.” Our charge in this
paper is to explain the very formation, design, and rationale of a “network plan” even before
the very first migrant has embarked on his voyage.
Just as a ROSCA is a means to overcome the lack of access to credit that is needed to
facilitate and expedite the purchase of a costly good in one’s locale, migration network is an
informal group-saving scheme aimed at facilitating and expediting access to a rewarding yet
costly employment opportunity in a location farther afield. However, migration networks
have an important feature distinct from the mechanisms of ROSCA as presented, for
example, by Besley et al. (1993), and Anderson et al. (2009). Namely, the enforcement of
future payments from a member of ROSCA who has won “the pot” early on depends on the
threat of social and material sanctions that other members are capable to impose. In
sustaining a ROSCA, a crucial factor is the physical and regular proximity of the members, a
feature that is absent in the context of migration. Put differently, whereas the study of
ROSCA is of a mechanism for arranging finances across time, the study of migration
network as a “dynamic” ROSCA is of a device for financing gainful activity both across time
and across space. Space matters because transactions are not seen by all members at
subsequent “meetings” (in each “meeting,” the number of members who are away increases
by one), and “collecting” from members who are far away is qualitatively distinct from
collecting from members nearby; direct and immediate enforcement devices available in the
latter case are not available in the former, for example. Put somewhat crudely, in the
spectrum spanned by the polar cases of spot exchanges and sequential exchanges, the
standard ROSCAs are placed significantly to the left of migration networks as dynamic
ROSCAs.
We present a setting in which in terms of utility-measured gains and opportunity
costs, a cooperation agreement will be preferred to “going it alone.” We show that the
agreement creates a network in which “established” migrants contract to support the
subsequent migration of others in exchange for being supported themselves, and that the
optimal size of the network (the number of the cooperating migrants) is finite.
2
See, for example, Banerjee (1983), Massey et al. (1987), and Munshi (2003).
3
Perceiving migration networks as mechanisms geared at financing and expediting
migration is not the only way of thinking about networks as a means of supporting and
facilitating follow-up migration. In earlier writings, we alluded to other variables and
mechanisms that explain why “established” migrants provide support for the follow-up
migration of others. These variables and mechanisms include: altruism (Stark, 1999); the
building up of a community of migrants to constitute a reference group that will constrain the
relative deprivation that would otherwise be felt through unavoidable comparisons with the
“natives” (Fan and Stark, 2007); wage gains (Stark and Wang, 2002); and the formation of a
political constituency (Stark, 1993). We also considered the support given to others as a
means of building up the individual’s reputation in the home community so as to cushion his
return (Lucas and Stark, 1985), although here we develop an argument premised on
permanent migration. The present perspective of networks adds to the received literature in a
number of concrete ways: it identifies a new rationale, both from the perspective of
established migrants and from the perspective of would-be migrants, for the prevalence of a
network; it considers membership in a network as a choice variable in an explicit
optimization process; it yields a precise prediction as to the timing and sequencing of
migratory moves by members of the network; it determines the optimal size (membership) of
the network; it establishes a link between the magnitude of remittances and the cost of
migration, and explicates the varying intensity of remittances over time; and it explains a
large number of stylized facts that were hitherto subject to a plethora of theories.
Before proceeding, we summarize the migration characteristics and stylized facts that
we seek to explain: 3
-
Migration is phased; migrants arrive at destination sequentially, not simultaneously.
-
Would-be migrants receive assistance from past migrants; past migrants provide
assistance to would-be migrants.
-
There are different levels of likelihood that migration will be mediated by networks,
depending, inter alia, on the cost of migration, and on the difference in earnings
between destination and origin.
-
Networks constitute an endogenously-generated voluntary arrangement, not an
exogenous pre-existing structure.
3
Several of the listed stylized facts are elicited from Stark (1993), Rosenzweig and Stark (1997), and Stark
(2009).
4
-
Networks are more likely to consist of individuals who are close to each other in the
community of origin (family members, friends) than of individuals who are little
related to each other at origin.
-
Tightly-linked communities in the home country are more likely to form larger
networks than communities with loose links.
-
Even in the case of relatively small communities, several networks may co-exist,
rather than all migrants and would-be migrants taking part in a single network.
-
Networks make it possible for individuals to migrate and / or to migrate earlier than
in the absence of networks.
-
Migrants remit.
In the next section we present an introductory analysis of two individuals, and we
show that in a complete-compliance cooperating scheme, the expected utility from
cooperation between the two individuals is always higher than the sum of the utilities from
saving for a migration alone, and that cooperation can enable migration even when going
alone does not. In section 3 we study a risk-laden cooperative scheme, alluding to the
perceived risk of taking part in an n-person cooperation agreement where the source of the
risk is the possibility that an individual who is randomly drawn to migrate will renege so as
to enjoy a higher level of utility sooner. We link the probability of reneging with the intensity
of interpersonal bonds (affinity) between the cooperating individuals. We find that there
exists a finite optimal size of the migration network. We show that the optimal size of the
network depends on the rate of decline in the affinity among the members as the size of the
network expands. Consequently, we infer that in a community a multitude of networks, rather
than a single grand network, will be formed. Section 4 sets out our conclusions.
2. A rewarding two-person cooperation aimed at expedited financing of
migration
Let y (t ) denote an individual’s flow of earnings, measured in income units (IU), in
continuous time t, measured in months, such that the individual receives
5
 y IU for working in the home country
y (t ) = 
Y IU for working in the foreign country,
 y IU for working in the home country
y (t ) = 
Y IU for working in the foreign country,
and where, to represent the fact that income in the destination country is higher than income
in the home country, Y > y > 0 .
The income of an individual can be divided into two components: consumption and
saving. 4 We denote the flow of savings of an individual in time t as s (t ) . Throughout, we
assume a zero rate of interest. 5
Let the individual’s utility function, u ( x(t )) , where x=
(t ) y (t ) − s (t ) is the
individual’s flow of consumption at time t, be a continuous and increasing function. The
intertemporal preferences of the individual are expressed by a continuous discount term e −δ t ,
where δ ∈ (0, +∞) is a discount factor, allowing us to write the utility experienced by an
individual during a lifetime lasting T months as
T
U ( x(t )) = ∫ e −δ t u ( x(t ))dt .
0
For simplicity, in the notation below we omit the argument x(t ) of the function U . The
lifetime utility of an individual who spends his entire lifetime in the home country, U H , is
T
U H = ∫ e − δ t u ( y )dt.
0
Case 1: A single individual (saving alone to facilitate migration)
Let the cost of migration to the destination country be equal to C IU, C > 0 . Consider an
individual who at the beginning of month t = 0 decides to save in order to migrate, denying
himself utility for a considerable period of time in order to enjoy later on a higher income
4
To concentrate on essentials, we assume that covering the cost of migration is the only reason to save;
individuals are not interested in “smoothing” their consumption over time or in other activities that depend on
intertemporal income (wealth) transfers.
5
We assume that in the home country capital markets and the banking sector are underdeveloped if not virtually
nonexistent, such that no institutionalized saving/credit possibilities are readily available to help pay for the cost
of migration. Moreover, even if a credit market were to exist, lenders will presumably be quite reluctant to
finance an “escape” of a borrower.
6
and utility in the destination country. For simplicity, we assume that in each month the
individual saves a constant amount out of his disposable income, a sum of s (t ) ≡ s A , which
translates into C / s A months of saving, where the subscript “A” stands for “alone.” (Of
course, because the saving period cannot be longer than the individual’s lifespan, we have
that s A ∈ [C / T , y ] .) Then, the lifetime utility of an individual who in order to migrate saves
alone at the rate s A , is equal to
=
U A (sA )
C / sA
∫
T
−δ t
e u ( y − s A )dt +
0
∫
e −δ t u (Y )dt.
(1)
C / sA
First, to render migration a possible option, the cost of migration must not be
exceedingly high in relation to the duration of the individual’s lifetime and to his earnings in
the home country. Specifically, the cost has to be lower than the lifetime income of the
individual when living in the home country, namely, C < yT .
Second, to render migration a rational choice for an individual, we assume that the
income in the destination country, Y, is sufficiently high to allow the individual to reap gains
from migration - compared to living in the home country - in the time span that remains after
saving for the migration trip. This requirement can be expressed as the condition
max U A ( s A ) > U H .
0≤ s A ≤ y
Considering U A as a function of the savings and of the income in the destination country,
U A ( s A , Y ) , let us denote by Y0 the level of income in the destination country that equalizes
(the optimal) consumption of the migrant and that of an individual living for his entire
lifetime in his home country, that is, Y0 is given implicitly by
max U A ( s A , Y0 ) = U H .
0≤ s A ≤ y
(2)
In order for migrating to constitute a gainful option for a single individual, we must then
have that Y > Y0 .
We refer to the saving rate, 0 ≤ s A ≤ y , that maximizes U A ( s A ) in (1) by s*A . Then,
saving at the rate s*A in order to accumulate the funds needed to pay for the cost of migration
amounts to TA = C / s*A months of saving, and therefore we can write the lifetime utility of an
individual who saves for migrating alone as
7
=
U A U=
A (s )
*
A
TA
T
0
TA
−δ t
*
−δ t
∫ e u ( y − sA )dt + ∫ e u (Y )dt.
Case 2: A cooperation agreement between two individuals (the formation of a migration
network)
We next consider an arrangement of two individuals in the home country joining forces and
saving together to cover the cost of migration and then send one of them to the destination
country, such that with his boosted income and savings the migrant will be able to help the
individual who stayed behind reach the destination country faster than had the latter saved
alone. The choice as to who of the two individuals will be the first-to-go migrant (also
referred to henceforth as the “winner”) is to be made by tossing a fair coin when the two
individuals between them have saved enough to meet the cost of migration by one of them. 6
Temporarily we assume a “complete-compliance” type of agreement, that is, we ignore the
possibility that the “winner,” tempted by the prospect to enjoy higher consumption sooner,
will not hold his part of the deal after he arrives at the destination country.
In such a scheme, it is possible, or even likely, that the “winner” will contribute more
to the common “pot of savings” after he departed, acting on his boosted income. To resolve
this “imbalance,” two scenarios then come to mind: in one scenario, the “loser” (the second
individual to go) will be required to repay what the “winner” advanced to the second
individual to support the latter’s migration. Proceeding in this way, the two-way financial
transfers are balanced, albeit the second individual is clearly worse-off in terms of the time
span earmarked for saving and holding back on enjoyment from consumption compared to
the first individual. In the second scenario, just as soon as the second individual arrives at the
destination country, both individuals start enjoying their higher earnings. This way, the
agreement between the two individuals is fairer in terms of the sacrifices that each of them
6
This scheme is somewhat similar to a “premium bidding ROSCA” described by Kovsted and Lyk-Jensen
(1999) in which individuals who are heterogeneous in their “business” skills compete for the pot by promising
higher contributions after they receive the accumulation in the pot, invest what they get, and reap high returns
from their investment. In such a ROSCA, the individual who can make the most from the investment is likely to
get the contents of pot the earliest. In our scheme, however, the participants migrate in random order because
they do not differ in their “migration productivities.”
8
makes. In the remainder of this paper we will assume the second scenario, which we consider
more appealing. 7
We assume that the individuals choose three optimal saving rates in order to
maximize their expected lifetime utility: these rates accrue when the individuals are both in
the home country (saving each at the common rate 0 ≤ sH 1 ≤ y because up to the moment of
the draw of who will be the first to migrate they are indistinguishable from each other,
accumulating between them 2 sH 1 IU per month), and when one individual is in the
destination country (saving 0 ≤ sD ≤ Y ) while the other is still at home (saving 0 ≤ sH 2 ≤ y ).
Then, the expected utility of an individual entering a “complete-compliance” two-person
cooperation scheme is equal to
1
1
EU Coop ( sH 1 , sH 2 , sD ) = UW ( sH 1 , sH 2 , sD ) + U L ( sH 1 , sH 2 , sD ) ,
2
2
(3)
where
U W ( sH 1 , sH 2 , sD ) =
C
2 sH 1
∫
e −δ t u ( y − sH 1 )dt +
C
C
+
2 sH 1 sD + sH 2
∫
C
2 sH 1
0
T
∫
e −δ t u (Y − sD )dt +
e −δ t u (Y )dt
C
C
+
2 sH 1 sD + sH 2
and
C
2 sH 1
C
C
+
2 sH 1 sD + sH 2
0
C
2 sH 1
−δ t
U L ( sH 1 , sH 2 , sD ) =
∫ e u ( y − sH 1 )dt +
∫
T
e −δ t u ( y − sH 2 )dt +
∫
e −δ t u (Y )dt
C
C
+
2 sH 1 sD + sH 2
are the lifetime utilities of, respectively, the “winner” and the “loser.”
Let us denote by sH* 1 , sH* 2 and sD* the saving rates that solve the maximization
problem
max
0 ≤ sH 1 ≤ y , 0 ≤ sH 2 ≤ y ,0 ≤ sD ≤Y
EU Coop ( sH 1 , sH 2 , sD ) ,
and by EU Coop = EU Coop ( sH* 1 , sH* 2 , sD* ) the optimal level of expected utility from the
cooperation. The following lemma formalizes the intuition that saving together is preferable
to saving alone.
7
Under joint utility maximization, when each individual receives an equal weight in the optimization and the
individuals have the same concave utility functions, it can easily be shown that this second arrangement is
optimal and hence preferable to the first.
9
Lemma 1: Migration in a complete-compliance cooperative agreement is always preferred
by a risk-neutral individual to saving alone, namely EU Coop > U A .
Proof: The proof is in the Appendix.
Another property of the complete-compliance cooperative agreement is that it can
make migration a gainful proposition even when it is not an appealing option for a single
individual due to the insufficient increase in the earnings in the destination country. Namely,
let us consider EU Coop as a function of the income in the destination country, EU Coop (Y ) , and
let us denote by Y1 the level of the earnings in the destination country for which
EU Coop (Y1 ) = U H .
Then we have the following lemma.
Lemma 2: The complete-compliance cooperative agreement lowers the minimal level of
income in the destination country that is necessary to render migration a rational choice
below the minimal level of income necessary to render saving alone a rational choice,
namely, Y1 < Y0 .
Proof: The proof is in the Appendix.
Summing together Lemmas 1 and 2, we can state that entering cooperation in order to
migrate strictly dominates saving alone in order to migrate and, moreover, entering
cooperation can render migration attractive even when the difference in earnings between
destination and home is muted.
Stated more forcefully, Lemma 1 informs that a decision to save alone can be
considered irrational. In the next section we consider, however, a plausible “dark side” of the
cooperation agreement - a cost that the loser could be exposed to upon a failure of the winner
to fulfill his part of the agreement. We introduce the possibility of cooperation by more than
two individuals and we show that the risk involved limits the optimal size of the cooperating
group.
In Stark and Jakubek (2012) we conduct a detailed analysis of the two-person
cooperation scheme. Drawing on a linear form of the utility function, we show that in a
complete-compliance scheme, a cooperation agreement decreases the opportunity cost of
migration, which is measured by the utility forfeited during the period of saving to pay for
the cost of migration. Then, introducing the risk arising from the possibility of the first-to-go
10
migrant defaulting after he gets to the destination country, we show that the propensity to
enter a risk-laden agreement increases with the cost of migration. Namely, as the cost of
migration increases, an individual will be willing to strike a cooperation agreement with a
counterpart whom he considers less reliable. The intuition behind this finding stems from the
fact that as the cost of migration increases, the gain from cooperation counters the possible
loss (sustained upon a counterpart failing to keep his side of the agreement) through two
channels. First, it can be shown that the benefit from successful cooperation, namely the time
gained in accumulating the requisite savings, is linearly increasing in C, as is the potential
cost of an unsuccessful cooperation, namely the time that a cheated individual loses. Second,
because the individual discounts future utility, a gain realized earlier due to cooperation
overshadows the possible pain to be sustained farther in the future.
3. The default risk, the inclination to enter cooperation, and the optimal
size of the cooperating group
In the preceding section, we saw that entering a cooperation agreement in order to meet the
cost of migration strictly dominates saving alone for this same purpose. However, because
the second-to-go migrant loses “control” over the first-to-go migrant after the latter’s
departure, there is a potential risk arising from the “imbalance of powers” between the two
individuals: should the first-to-go migrant decide to renege and enjoy higher utility by
neglecting to remit to help the loser of the draw to migrate, the latter might have little in the
way of reacting. It then stands to reason that an individual will be reluctant to strike
cooperation with a “random” individual or a “stranger;” instead, he will prefer a counterpart
whom he knows and whom he considers sufficiently reliable so as to render the default risk
bearable. 8
We study a setting that involves possible cooperation between more than two
individuals. The questions that we seek to address are as follows: if there are conditions
under which cooperation in saving for migration and a phased schedule of departures by two
individuals dominate saving alone, then under what conditions will there be pooling of
8
The issue of reliability or of the confidence placed in the migrant being a determining factor of the choice of a
migrant is not a novelty unraveled by this paper. Nearly a quarter of a century ago it was argued that families in
the Philippines select a daughter rather than a son as a migrant even though the earnings of a son as a migrant
are expected to be higher, and that this selection is made because daughters are considered to be more reliable
remitters (Lauby and Stark, 1988).
11
savings by n ≥ 2 individuals? What are the characteristics of a group scheme? In particular,
is there an optimal group size? And if so, what determines or binds the size of the group?
In their study of Mexican migration to the U.S., Massey et al. (1987) catalogue
intensities of affinity, including “most important kin relationships in migrant networks
[which] are those between fathers and sons, uncles and nephews, brothers, and male
cousins,” weaker links that are based on friendship or paisanaje (a common community of
origin), and so on. To quantify the degree of affinity between individuals, we assume that the
population of the home country is countably infinite, and that the bond or affinity between a
pair of individuals is measured by a single value that ranges between zero and one. The
values of the affinities of individual j ( j = 1, 2,... ) towards individuals i = 1, 2,... are given by
a sequence P j = ( p1j , p2j ,...) , where 0 ≤ pij ≤ 1 for i = 1, 2,... . Because affinity is mutual, we
presume that pij = p ij .
To ease the analysis, we sort the values of affinity such that the sequence P j is nonincreasing for each j; that is, an increase in the index i yields pij values that correspond to
individuals who are increasingly less related to j. 9 For example, members of j’s family will
be accorded the highest pij values and be numbered by the lowest i ’s, closest friends a little
lower pij ’s and a little higher i ’s, and so on. Furthermore, the individual bears no affinity
toward individuals who are exceedingly removed from him. Thus, for every individual
j = 1, 2,... we have that p1j = 1 ; pij ≥ pij+1 for i = 1, 2,3,... ; and lim pij = 0 .
i →∞
To calculate the expected utility from cooperation, we assume that individual j
interprets a pij value as the probability that the counterpart i fulfills his part of the sequential
financing agreement when i is the winner of the draw. The one-to-one mapping between
affinity and the probability of migrant i keeping the agreement is premised on the notion that
it is hard for an individual to cheat someone who is close to him. 10 Individual j will then
compare his expected lifetime utility from cooperation with his lifelong utility from saving
alone. Being risk-neutral, he will prefer cooperative saving to saving alone when the
9
Strictly speaking, as the ordering of the index i is now different for each j-th individual, we should have
denoted the indices as i ( j ) , but because in what follows we evaluate the gains from migration only from the
perspective of a single j-th individual, we elected not to clutter the notation for no discernible gain.
10
Although we do not model this possibility explicitly, another explanation for the presumption that an
individual finds it hard to cheat someone who is close to him is that the individual is likely to harbor altruistic
feelings towards those close to him, making him enjoy to some extent the greater pleasure that they obtain from
a successful cooperation or, conversely, rendering cheating them harder because this would decrease his utility.
12
expected utility from cooperation is higher than the utility from saving alone. Because in
what follows we evaluate the gains from migration only from the perspective of a single
individual, the superscript j is dropped.
We consider a setting in which n individuals, n ≥ 1 , 11 save together in order to
expedite the migration of the group, and we prove a general property of such a saving
scheme, namely the existence of an optimal size of the migration network.
As before, we assume that the choice as to who is the first to go, who is the second to
go, and so on, is made by means of a random draw at each point in time when the group
happens to accumulate enough savings to pay for the migration of one member. 12 Therefore,
when evaluating ex-ante the gains from a cooperation agreement, an individual uses the mean
value of the affinity across the potential participants, pn =
1 n
∑ pi , as the probability of
n − 1 i =2
cooperation being successful at each k-th step, k = 1,..., n , in the case in which he is not
drawn as the k-th-to-go migrant, an event that occurs with probability
n − (k − 1) − 1 13
.
n − (k − 1)
Assuming a “once beaten, twice shy” type of behavior, the cooperation scheme collapses
when the first “deviator” appears among those who already made the trip, and each of the
cheated individuals who are still in the home country will then, in order to migrate, start to
save on his own, or, if at that point in time migration facilitated by saving alone is no longer
attractive, he will forfeit saving and spend his income in the home country.
Thus, the expected utility of an individual who in a group of n individuals enters an
agreement is
1
n −1 
n−2
 1

EU n = U n :1 +
1 − pn ) U n :1,Ch + pn 
U n :2 +
1 − pn ) U n :2,Ch + pn ... U n :n      ,
(
(



n
n 
n −1
 n −1

where U n :k is the utility of an individual who was drawn as k-th-to-go, and U n :k ,Ch is the
11
For the sake of completeness, we incorporate the case of an individual who saves all by himself.
Alternatively, the order of taking the migration trip could be determined through votings by the group
members who, at each round, select the individual who is collectively most trusted, namely, the individual for
whom the sum of the affinities is the highest. However, because no individual possess complete information on
the affinity levels of others, the outcome of the voting is not known to the “representative” individual ex-ante.
Specifically, the sequence to be chosen by the group will likely be different from the sequence that the
“representative” individual will consider optimal. Therefore, in such a case too, the expected gain from a
cooperation agreement, as calculated by a “representative” individual, will depend on the mean value of the
affinity taken over the members of the group.
12
13
For the sake of notational consistency, for n = 1 we set p1 = 1 .
13
utility of an individual betrayed at step k. That is, at step k = 1 a reference individual faces a
probability 1/ n of being the winner of the draw, and a probability (n − 1) / n of ending up as
the loser of the draw, in which case his “fate” depends on the behavior of the winner whose
propensity to fulfill the agreement is pn . If the winner does not renege, then at step k = 2 the
reference individual has a chance to be drawn, this time with probability 1/ (n − 1) , or he still
stays at home, which happens with probability (n − 2) / (n − 1) , such that his “fate” depends
on the second-to-go migrant whose propensity to fulfill the agreement is pn . And so on. We
do not specify the explicit expressions for U n :k ,Ch and U n :k , because these utilities depend on
the saving rates chosen by the cooperating individuals and, furthermore, U n :k also depends
on the behavior of the reference individual (whether he will choose to renege when drawn as
k-th-to-go) and on the behavior of those drawn as the next-to-go after he was drawn (in case
one of them reneges, the scheme collapses and the promise to contribute to the group savings
is no longer binding).
Lastly, and as before, we assume that the income in the destination country is high
enough as to make a single individual willing to migrate, namely, as already stated following
(2), that Y > Y0 .
We are interested in the gain from entering a cooperation agreement compared to
saving alone, namely in the difference
∆=
U n EU n − U A ,
presuming that if ∆U n > 0 , a risk neutral individual will prefer cooperation in a group of size
n to saving alone. We now state and prove the following claim.
Claim 1: There exists an l ≥ 1 such that ∆U l = max {∆U n } . Also, the set { j : ∆U j = ∆U l }
n =1,2,3,...
has a minimum, to which we refer as the optimal membership of the cooperation agreement
(the optimal size of the migration network).
Proof: The proof is in the Appendix.
Claim 1 establishes the main result of this section: there exists a finite optimal size of
a group of cooperating migrants. Our model predicts that in order to facilitate migration,
individuals in the home country will elect to “cluster” in separate groups consisting of
members linked by interpersonal ties, a feature that allows for sufficient mutual confidence.
14
Put differently, our model predicts formation of networks of limited size rather than the
formation of a single grand network. As indicated by the proof of Claim 1, the optimal size
of the network depends on the interplay between the potential increase of the U n :k terms
(describing the utility of an individual who participates in the n-individual cooperative
arrangement) and the rate of decrease of pn (the affinity that characterizes the n-th individual
who is included in the cooperation agreement) as n grows. Since the U n :k values are similar
for poor countries of the same earnings gap with a given country of destination, the extent to
which the groups will comprise of members of specific families, clans, villages, and so on
will depend on characteristics of the home country population, as displayed by the
distribution of P. In particular, from the proof of Claim 1 it follows that if the links between
members of a population are strong - a characterization represented by pn values (and
likewise by the value of pn ) fading to zero slowly (c.f. (A5)) - the optimal size of a network
is likely to be large, because the sequence (∆U1 , ∆U 2 , ∆U 3 ,...) will also decrease slowly.
Conversely, if the sequence pn drops to zero fast - strong bonds in a population are confined
to the relatively small circle of the family and closest friends - the networks will be limited in
size, and in the case of exceptionally “bonds-less” population, going-alone will be the only
option.
4. Conclusions
We modeled the formation of a migration network, viewing the network as an informal
financial cooperation between (would-be) migrants, intended to facilitate and expedite their
departure to the destination country and enjoy there higher earnings. We showed how
intertemporal cooperation can substitute for intertemporal borrowing, and how it can help
avoid several of the drawbacks of uncollateralized borrowing such as high interest rates. Our
analysis showed how in the presence of commitment, would-be migrants can benefit from
jointly financing each other’s cost of migration, and that the size of the migration networks is
bounded. The arrangement described suggests that it is not necessary for a would-be migrant
to either accumulate savings solely from his own home country income, or to become
indebted to traffickers. Other considerations being the same, the more likely it is that the
conditions exist for striking the kind of cooperative arrangement that we have outlined, the
less likely it is that such exploitative organizations will be called upon to facilitate migration.
15
With an agreement of the type stipulated in our model, migration will be sequential:
without an agreement, migration will be simultaneous. The possibility of distributing the
departure points over time gives rise to a migration network. Put differently, the possibility of
striking an agreement generates a network, and a network constitutes evidence that an
agreement has been struck. Thus, cooperation, networks, and sequencing are interlinked. The
model of group migration presented in Section 3 stipulates an expedited migration flow that
ceases when the last participant in the cooperation agreement ends up migrating. This
depiction aligns with a finding of Hanson and McIntosh (2010, p. 807) “[that] the networks
created by labor supply-driven migration are self-reinforcing over time only if those
networks are new; [Mexican] states in which those networks were already extant show a
dampening, rather than an acceleration, over time.”
A cost-based rationale for the formation and functioning of a network is clearly not
the only possible rationale; as noted in research on the motivation of migrants to remit,
motives can range from pure altruism to pure self-interest (Lucas and Stark, 1985; Stark and
Lucas, 1988; Stark, 2009). Here, we have explored one of the options motivated by selfinterest.
16
Appendix
Proof of Lemma 1
We consider a saving plan in which, when at home, the two individuals save at the same rate
as an optimizing single migrant-to-be, that is, s=
s=
s*A , whereas from the time the firstH1
H2
to-go migrant is at the destination country, he retains the consumption level he had in the
home country, thereby saves the entire income increment he has in addition to the rate s*A ,
that is, sD = s*A + Y − y . Clearly, because Y > y , we have that sD > s*A . Obviously, such a
saving plan is feasible, and therefore we have that
=
EU Coop EU Coop ( sH* 1 , sH* 2 , sD* ) ≥ EU Coop ( s*A , s*A , s*A + Y − y ) .
(A1)
Then, the time needed to save for the two migration trips is shorter than the time it
takes a single individual to be able to migrate, as
C
C
C
+ *
< *
*
2sA 2sA + Y − y s A
for Y > y . Thus, comparing EU Coop ( s*A , s*A , s*A + Y − y ) with the maximum utility level
available to a single migrant, U A ( s*A ) , we have that
1
1
UW ( s*A , s*A , s*A + Y − y ) + U L ( s*A , s*A , s*A + Y − y )
2
2
EU Coop ( s*A , s*A ,=
s*A + Y − y )
C
2 s*A
∫e
−δ t
u ( y − s )dt +
C
C
+
2 s*A 2 s*A +Y − y
∫
*
A
T
−δ t
e u ( y − s )dt +
*
A
C
0
=
∫
(A2)
T
−δ t
e u ( y − s )dt +
*
A
C
0
∫
−δ t
e u (Y )dt
C
+
2 s*A 2 s*A +Y − y
C
>
s*A
T
−δ t
*
−δ t
U A ( s*A ).
∫ e u ( y − sA )dt + ∫ e u (Y )dt =
0
e −δ t u (Y )dt
C
C
+
2 s*A 2 s*A +Y − y
2 s*A
C
C
+
2 s*A 2 s*A +Y − y
∫
C
s*A
Joining (A1) and (A2) we get that EU Coop > U A . □
17
Proof of Lemma 2
Let
GA (Y ) = max U A ( s A , Y ) − U H
0≤ s A ≤ y
and let
GCoop (Y ) = EU Coop (Y ) − U H .
Recalling the definitions of Y0 and of Y1 , we have that GA (Y0 ) = 0 , and that GCoop (Y1 ) = 0 .
Obviously, GA (Y ) and GCoop (Y ) are increasing in Y, a fact that together with the inequality
GA (Y ) < GCoop (Y ) which is implied by Lemma 1, translates into Y1 < Y0 . □
Proof of Claim 1
Clearly, in case of a single individual,
EU1 = U A ,
so
∆U1 = 0 .
(A3)
For the case n > 1 , let us denote by U k >1 the part in the equation for EU n which
describes the expected utility after the first-to-go migrant turned out to be honest, that is, let
1
n −1
(1 − pn ) U n :1,Ch + pnU k >1  .
EU n = U n :1 +
n
n 
We have that
U n :k < U Dest
and that
U k >1 < U Dest ,
T
for every k = 1,..., n , where U Dest = ∫ e −δ t u (Y )dt < ∞ , as surely the level of utility available to
0
the migrant is lower than the level achievable upon his entire lifetime hypothetically being
18
spent in the destination country, enjoying there the highest level of consumption.
Additionally,
U n :k ,Ch < U A
because a cheated individual has to either start saving from scratch being already “delayed”
by unsuccessful group saving, or he chooses to stay in the home country, in which case (c.f.
(2)) his utility will be lower than that of an individual who saves alone for migrating.
In consequence, for n ≥ 2 we have that
∆U n = EU n − U A =
1
n −1
(1 − pn ) U n :1,Ch + pnU k >1  − U A
U n :1 +
n
n 
n −1
1
< U Dest +
→U A − U A = 0,
(1 − pn ) U A + pnU Dest  − U A 
n →∞
n
n 
(A4)
because
1
U Dest 
→0
n →∞
n
and because
=
=
pn lim
pn 0 ,
lim
n →∞
which obtains because pn =
n →∞
(A5)
1 n
pi is a Cesáro mean of the sequence ( p2 , p3 ,...) , 14 we
∑
n − 1 i =2
have that
n −1
→U A .
(1 − pn ) U A + pnU Dest  
n →∞
n 
Consequently, the sequence (∆U1 , ∆U 2 , ∆U 3 ,...) starts at zero (c.f. (A3)) and is
bounded from above by a sequence that has a limit equal to zero for n → ∞ (c.f. (A4)).
Therefore, one of the following cases is true.
∆U1 = max {∆U n } ; group cooperation is not a viable option (the optimal size of
1. 0 =
n =1,2,3,...
the “network” is equal to one).
14
The Cesáro mean of a sequence ( a1 , a2 ,...) is a sequence (c1 , c2 ,...) such that ci = ( a1 + a2 + ... + ai ) / i . It is
easy to see that if lim an = A then also lim cn = A .
n→∞
n→∞
19
2. There exist j > 1 such that ∆U j = max {∆U n } , and ∆U j > 0 , and a number
n =1,2,3,...
=
l min{
=
i : ∆U i ∆U j } , l > 1 , to which we refer as the optimal size of the migration
network. □
20
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